Corollary 3.4
Suppose that
X
∈
IR
l
is starshaped with respect to
0
and locally polyhedral at
0
,
0
∈
X
and
p
∈
*
IR
l
.
Then
pX
≥
0
implies
p
(
*
X
)
≥
0
.
Proof.
Let
P
be a polyhedron obtained as an intersection of
X
with a suffi
ciently small closed cube, whose interior contains 0
.
Suppose that there exists
˜
x
∈
*
X
such that
p
˜
x <
0
.
Then one can find ˜
x
0
=
ε
˜
x
∈
*
P
such that
p
˜
x
0
<
0
,
for some sufficiently small
ε
∈
*
IR
++
.
But
pP
≥
0
,
which by Lemma 3.3 implies
p
(
*
P
)
≥
0
,
a contradiction.
2
Proof of Proposition 3.2.
Let
q
p
= (
q
1
, . . . , q
k
) for some
k
≤
l.
If
p
= 0 then
q
1
= 0
, k
= 1 and
¯
B
(
p
) =
X
=
B
1
(
p
)
.
Suppose that there exist a number
m
∈
{
1
, . . . , k
}
and an element
y
∈
X
(
q
1
, . . . , q
m

1
) such that
q
m
y <
0
.
Moreover,
assume that
m
is the smallest such a number, which guarantees that
q
j
X
(
q
1
, . . . , q
j

1
)
≥
0
,
j
∈ {
1
, . . . , m

1
}
.
(7)
Note that by construction
y
∈
B
m
(
p
)
.
If
m
= 1 then
¯
B
(
p
) =
B
(
q
1
) =
B
1
(
p
) by
virtue of Proposition 2
.
4
.
Assume
m
∈ {
2
, . . . , k
}
and show that
¯
B
(
p
) =
B
m
(
p
)
.
First, we shall prove that
¯
B
(
p
)
⊆
B
m
(
p
)
.
Take some arbitrary
x
∈
¯
B
(
p
) and
suppose that
x
6∈
B
m
(
p
)
.
If so, then by the choice of
m
and the system of
inequalities (7) there exists a number
j
∈ {
1
, . . . , m
}
such that
q
j
x >
0
,
q
t
x
= 0
,
t
= 1
, . . . , j

1
.
(8)
Consider some arbitrary nonstandard element ˜
x
∈
*
X
such that ˜
x
≈
x.
Then
q
j
x >
0 implies that
q
j
˜
x
is greater than some strictly positive real number.
Take now any standard
x
0
∈
X
and consider the first nonzero element in the
ordered set
{
q
1
x
0
, . . . , q
j

1
x
0
}
.
Since
j
≤
m,
it follows again from (7) and the
choice of
m
that such an element, if it exists, is strictly positive. Therefore, a
nonstandard linear functional
λ
1
q
1
+
· · ·
+
λ
j

1
q
j

1
takes only positive values
on
X
:
‡
λ
1
q
1
+
· · ·
+
λ
j

1
q
j

1
·
X
≥
0
.
Then by Lemma 3.3
‡
λ
1
q
1
+
· · ·
+
λ
j

1
q
j

1
·
*
X
≥
0
.
(9)
Consider
1
λ
j
p
˜
x
=
1
λ
j
[
λ
1
q
1
+
· · ·
+
λ
j

1
q
j

1
]˜
x
+
q
j
˜
x
+
1
λ
j
[
λ
j
+1
q
j
+1
+
· · ·
+
λ
k
q
k
]˜
x.
13